The Universe Revealed

Crazy Quantum Physics

Crazy Quantum Physics

It's not really crazy but it is interesting. Basically, I want to delve into explanations for all of the strange terms you hear when it comes to quantum physics.

First of all, much of the math involves vector spaces. A vector space is a collection of two or more vectors. You have a vector on the x-axis and one on the y-axis and then you combine them and get a new vector that is between the two original vectors. The combination of the new vector and the old ones are considered a vector space. You can also have vector spaces in three dimensions, and in fact that's the usual case. Euclidean vector space is in three dimensions. In other words, a vector has x, y, and z coordinates in this case.

A Tensor is a geometric object that involves vectors spaces.

A Tensor product is what you get when you multiply two vector spaces together. The product of these two vector spaces is a Tensor.

Well, it seems that this wasn't good enough so David Hilbert came up with Hilbert space. This is where the vectors are in more than three dimensions, and in fact can be in an infinite number of dimensions. All of this stems from linear algebra in which vectors have an infinite list of coordinates that can be added together, and the angles between vectors and the lengths of these product vectors are finite. In other words, the products of two vectors are not infinite.

Hilbert deals with complete metric spaces in which there are no missing points inside or at the borders. A Hilbert space is a real or complex inner product space, which means that it's a vector space with a linear product, which links each pair of vectors with a scalar (measurable number) quantity. In the case of a Hilbert space this linear product has length and angle. When the Hilbert space is complex then it has an inner product of the two vectors that has a complex number with each pair of elements x, y of Hilbert space that satisfies three properties.

The possible pure states of a quantum system (think particle here) are typically represented by unit (state) vectors that reside in a complex Hilbert space, and these possible states are points that reside in the complex space. In other words, there are an infinite number of possibilities for the states of a quantum system.
Let's look at it this way. In quantum physics, probabilities take the place of absolute measurements. What you start with is a column of vector probabilities. It's like as if you threw a dice and wrote down the results in a column. This would represent the vector probabilities of the possible states the particle could exist in (in this case a dice). However, quantum physics doesn't deal with probabilities but rather with probability amplitudes so you take the square roots of the probabilities. To find the actual probability that a particle will exist in a definite state you add the wave functions (the squares) and then take the square root of the result to get the summed vector. Then you take this summed vector and divide each probability (the square roots) by it to obtain the probability of a specific state.
Now it gets a bit more complicated.

Operators in quantum physics are used as mathematical rules. First you have to understand that there is a bra and a ket type of notation (this was invented by Paul Dirac). A bra (a list of vectors in a row) is the Hermitian conjugate of the corresponding ket (a list of vectors in a column). The reason why they do this is to avoid writing a complicated matrix by reducing it to a much simpler looking notation. An example of writing a matrix of vectors in ket notation is as follows: | < Ψ|ϕ >|^2 ≤ < ψ}ψ >< ϕ|ϕ >  This means that the square of the absolute value of the product of two state vectors, | < Ψ|ϕ >|^2 , is less than or equal to < ψ}ψ >< ϕ|ϕ >. See what I mean about being complicated.

What are the physical states in quantum physics? If you consider quantum particles, one of these states is spin. It turns out that particles don't actually spin like you would imagine (at one time they were considered to spin on an axis). What this means in quantum physics is an intrinsic angular momentum that is a vector with magnitude and direction, but with the intrinsic value of 0, + or - 1 or 1/2, and these are, in fact, considered a quantum number, which means that all particles of the same type have the same spin quantum number.

Quantum numbers are unlike normal numbers. They indicate states, which are discrete.
In essence quantum physics considers a physical system (particles) as having three basic things: states, observables and dynamics. States are vectors in Hilbert space. An observable is a gauge of a property of a system state that can be determined by a sequence of physical operations sort of like measuring the weight of something by putting it on a scale. In quantum physics of a particle this would be a mathematical operation on wave functions. Dynamics involve how changes occur with time. An example of this is the Schrödinger equation, which has partial derivatives with respect to time.

This only scratches the surface of quantum physics. I wrote this to show how complex this subject is and how physicists are approaching it. It's not easy to understand, but now you know why.

Thanks for reading.